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This document outlines five theorems that can prove if a quadrilateral is a parallelogram: if both pairs of opposite sides are congruent; both pairs of opposite angles are congruent; an angle is supplementary to both of its consecutive angles; the diagonals bisect each other; or one pair of opposite sides are congruent and parallel. It also summarizes the key properties of a parallelogram - both pairs of opposite sides are parallel and congruent, both pairs of opposite angles are congruent, or the diagonals bisect each other.
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This document discusses two postulates for proving triangles congruent: the Side-Side-Side (SSS) postulate and the Side-Angle-Side (SAS) postulate. It provides examples of proofs using each postulate, demonstrating how to prove triangles congruent by showing that three sides or two sides and the included angle are congruent between the two triangles. The document includes practice problems applying these postulate proofs.
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This document discusses similarity and congruency of triangles. It defines similarity as two polygons having equal corresponding angles and proportional corresponding sides. Congruency requires equal corresponding sides and angles. The document outlines three criteria for similarity: AAA, SSS, and SAS. It also covers the basic proportionality theorem, areas of similar triangles being proportional, and how Pythagoras' theorem relates to right triangles being similar.
Properties on Parallelograms Grade 9.pptx
The document presents 5 theorems about parallelograms: 1) opposite sides are congruent, 2) opposite angles are congruent, 3) consecutive angles are supplementary, 4) diagonals bisect each other implies the quadrilateral is a parallelogram, 5) diagonals form two congruent triangles. Each theorem is accompanied by an example proof using statements and reasons.
Congruence of triangle
The document discusses different types of triangles and rules for determining if triangles are congruent. It defines congruent figures as those that are equal in size and shape and can cover each other completely. It then provides four rules for triangle congruence: SAS, ASA, SSS, and RHS. These rules state that triangles are congruent if their corresponding sides and/or angles are equal.
Triangles
Triangles can be classified based on side lengths as equilateral, isosceles, or scalene triangles. They can also be classified based on angles as acute, obtuse, or right triangles. The three main properties of triangles are the angle sum property, exterior angle property, and Pythagorean theorem. Congruent triangles can be proven using the SSS, SAS, ASA, AAS, or RHS criteria which compare corresponding sides and angles. Important parts of a triangle include the median, altitude, angle bisector, and perpendicular bisector. The triangle inequality states that the sum of any two sides must be greater than the third side. Important points related to triangles are the incenter, circumcenter,
Math's ppt on triangles
Triangles can be classified based on side lengths as equilateral, isosceles, or scalene triangles and based on angles as acute, obtuse, or right triangles. The three main properties of triangles are the angle sum property, exterior angle property, and Pythagorean theorem. Secondary parts of a triangle include the median, altitude, perpendicular bisector, and angle bisector. Triangles can be proven congruent using the SSS, SAS, ASA, AAS, or RHS criteria. Inequalities in triangles relate longer sides to larger angles and shorter sides to smaller angles. Important centers of a triangle include the incenter, circumcenter, centroid, and orthocenter.
Geo 4.2 3-notes_cong_triangles
The document discusses four postulates and theorems for determining if two triangles are congruent: the Side-Side-Side postulate, the Side-Angle-Side postulate, the Angle-Side-Angle postulate, and the Angle-Angle-Side theorem. These state that if corresponding sides or angles of two triangles are congruent in certain combinations, then the two triangles are congruent. Homework assigned is to complete problems #25 from the 4.2 and 4.3 worksheet.
3-6 Congruent Angles
This document discusses angle relationships and congruence. It defines congruent angles as angles that have the same degree of measure. The Vertical Angles Theorem states that vertical angles are congruent. Several theorems are also presented: if two angles are congruent, then their complements and supplements are congruent; if two angles are complementary or supplementary to the same angle, then they are congruent; and if two angles are congruent and supplementary, then each is a right angle. The document provides examples to solve for the measure of unknown angles using these theorems and properties. Homework problems are assigned from pages 126-127, problems 9-21 and 24-28.
Geom 6point2 97
The document discusses properties of parallelograms including: opposite sides are congruent, opposite angles are congruent, and consecutive angles are supplementary. It also states that the diagonals of a parallelogram bisect each other. Examples are provided to demonstrate using these properties to find lengths, angle measures, and prove statements about parallelograms.
All about triangles (2)
This document discusses triangle similarity and how to determine if triangles are similar. It defines triangle similarity as when triangles only differ in size, with all angles being equal and corresponding sides having the same ratio. It then outlines several theorems - SAS, SSS, AAS, AA - that can be used to prove triangles are similar by comparing their sides and angles. Examples are provided to illustrate each theorem, and an interactive model is referenced for students to explore triangle similarity on their own.
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“Determine the condition that make a quadrilateral a parallelogram.”.pptx
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10.1 terms
The document defines key terms related to circles and tangents, including: a circle as points equidistant from a center point; a radius and diameter; a chord as a segment with endpoints on the circle; a secant as a line intersecting a circle at two points; and a tangent as a line intersecting a circle at exactly one point. It states that a tangent line is perpendicular to the radius drawn to the point of tangency, and that for two segments from the same exterior point to both be tangent to a circle, they must be congruent.
9.1 terms
Triangles formed by drawing an altitude to the hypotenuse of a right triangle are similar to the original triangle and each other. The altitudes of a right triangle act as geometric means for the similar triangles, with the leg altitudes being geometric means between the close short segment and full base, and the right angle altitude being the geometric mean between the two parts it creates on the hypotenuse.
6.1 terms
This document defines and describes various types of polygons including: convex polygons which have no interior angles, concave polygons which have interior angles, equilateral polygons with all sides congruent, equiangular polygons with all interior angles congruent, and regular polygons which are both equilateral and equiangular. It also defines vertices, diagonals, and explains the interior angle sum theorem for quadrilaterals.
8.6 terms
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11.2 terms
This document discusses how to calculate the areas of regular polygons such as equilateral triangles and other shapes. It explains that the area of an equilateral triangle is (side^2) * √3/4 and the area of a regular polygon is (1/2) * apothem * perimeter. The apothem is the distance from any side to the center and the central angle is found by dividing 360 by the number of sides.
8.2 terms
This document discusses proportions and their properties in geometry. A proportion is two equal ratios that can be set up and cross-multiplied to solve for unknown values. The terms of a proportion can be switched diagonally or by adding denominators to numerators while maintaining equality. The geometric mean is the value diagonal to itself in a proportion of two positive numbers.
6.7 terms
This document discusses the formulas for calculating the areas of various shapes. It states that congruent polygons have the same area, and the area of a region is the sum of non-overlapping areas. It then provides the area formulas for a square, rectangle, parallelogram, triangle, trapezoid, kite, and rhombus. For a rhombus, it notes that the area can also be calculated as for a parallelogram using base times height.
6.4 terms
This document defines and compares properties of rhombuses, rectangles, and squares. A rhombus is a parallelogram with 4 congruent sides and perpendicular diagonals. A rectangle is a parallelogram with 4 right angles and congruent diagonals. A square has the combined properties of a rhombus, rectangle, and parallelogram - it is a rhombus and rectangle with 4 equal sides and right angles.
10.4 terms
If lines intersect within or on the edge of a circle, the measure of the angles formed can be determined based on the intercepted arcs. For intersections inside the circle, the angle measure is half the sum of the intercepted arcs. For intersections on the edge of the circle, the angle measure is half the intercepted arc. For intersections outside the circle, the angle measure is half the difference of the intercepted arcs.
10.2 terms
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11.5 terms
The document discusses calculating the areas of circles and sectors. It states that the area of a circle is calculated as A = πr2, and the area of a sector is calculated as (Arc Measure/360)(area of the full circle), which finds the percentage of the total circular area that the sector represents by dividing the arc measure in degrees by 360.
11.1 terms
The document discusses angle measures in polygons. It states that the sum of interior angles in any polygon is (n-2)180, where n is the number of sides. It also explains that the sum of exterior angles is always 360, and each exterior angle is calculated as 360/n, where n is again the number of sides.
10.3 terms
This document discusses inscribed angles and their relationship to circles. It states that an inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. It also notes that an inscribed angle measures half the degrees of its intercepted arc and that two inscribed angles intercepting the same arc together measure the full arc. Additionally, it mentions that if a right triangle is inscribed in a circle, its hypotenuse is the diameter of the circle and that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
9.5 terms
This document provides information about trigonometric ratios and using a calculator to evaluate trigonometric functions. It defines the sine, cosine, and tangent ratios using the SOHCAHTOA mnemonic. It also explains that calculators may require typing the trig function before or after the number, and some may not require hitting the equals sign, so it is important to know how to use the specific calculator.
10.6 terms
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9.2 terms
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8.1 terms
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6.6 terms
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11.3 terms
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6.3 terms
- 1. Terms of Endearment Proving Quadrilaterals are Parallelograms: 6.3
- 2. Theorems that prove quadrilaterals to be parallelograms • If both pairs of opposite sides are congruent, it is a parallelogram • If both pairs of opposite angles are congruent, it is a parallelogram • If an angle is supplementary to both of its consecutive angles, it is a parallelogram • If the diagonals bisect each other, it is a parallelogram • If one pair of opposite sides are congruent and parallel, it is a parallelogram
- 3. Summary of properties that prove it a parallelogram • Both pairs of opposite sides are parallel • Both pairs of opposite sides are congruent • Both pairs of opposite angles are congruent • One set of sides is congruent and parallel • Diagonals bisect each other • One angle is supplementary to both of its consecutive angles